The Circular Aperture: Axial Modes
The circular aperture was used in conjunction with the piezoelectric device to observe axial and transverse modes of the HeNe laser. The digital oscilloscope allowed me to observe the modes as they appeared and disappeared. The aperture was adjusted so that it allowed only the TEM00 mode to pass through it. At this point, axial modes were present. Images from the digital oscilloscope were captured using a computer for three different cavity lengths: 27.5 cm, 40.5 cm, 56.1 cm. The images are shown below for each of the three cavity lengths.
The graphs shows two axial modes of one Doppler envelope followed by all three modes of another envelope. The distance between two Doppler envelopes, the FSR, is known to be 2 GHz. Knowing the vaule of the free spectral range of the laser and the relative distance between two envelopes and two adjacent modes in a Doppler envelope, the length of the laser cavity may now be calculated. To perform this calculation, we use the following formula for the frequency difference between adjacent modes:
This equation derives from the fact that:
where n equals the number of half wavelengths contained in the laser cavity. By combining these two equations we discover that:
A mode produced by n half-wavelengths yields the equation:
while an adjacent mode produced by n +1 half-wavelengths yields the expression:
The difference between the two frequencies is:
which simplifies to the equation shown above for the the frequancy difference:
Using this equation and the frequency difference between adjacent modes on the graph, 565,641,509 Hz, we calculate a cavity length of 26.5 cm. This figure has a percentage error of 4% compared to the measured value of 27.5 cm.
The same can be done for an intermediate cavity length of 40.5 cm. The graph below shows two Doppler envelopes, each containing three modes. The frequency difference between adjacent modes should be less in this case because according to our equation, as L increases the frequency difference will decrease. This is indeed the case. The frequency difference here is 390,860,495 Hz. The cavity length is calculated to be 38.35 cm. The theoretical value has a 5% error compared to the measured cavity length.
Finally, a calculation of the laser cavity was done for a long cavity of 56.1 cm. In this case there are four modes in each envelope because the frequency difference has decreased with increased cavity length such that an additional mode can now fit inside the envelope. The frequency difference between modes for the long cavity is 268,148,479 Hz. The cavity length is calculated to be 55.9 cm yielding a percent error of less than 1%.